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Theorem rdgon 5973
 Description: Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.)
Hypotheses
Ref Expression
rdgon.1 (𝜑𝐹 Fn V)
rdgon.2 (𝜑𝐴 ∈ On)
rdgon.3 (𝜑 → ∀𝑥 ∈ On (𝐹𝑥) ∈ On)
Assertion
Ref Expression
rdgon ((𝜑𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝜑,𝑥
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rdgon
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5178 . . . . 5 (𝑧 = 𝑥 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘𝑥))
21eleq1d 2106 . . . 4 (𝑧 = 𝑥 → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
32imbi2d 219 . . 3 (𝑧 = 𝑥 → ((𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On) ↔ (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On)))
4 fveq2 5178 . . . . 5 (𝑧 = 𝐵 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘𝐵))
54eleq1d 2106 . . . 4 (𝑧 = 𝐵 → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (rec(𝐹, 𝐴)‘𝐵) ∈ On))
65imbi2d 219 . . 3 (𝑧 = 𝐵 → ((𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On) ↔ (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ∈ On)))
7 r19.21v 2396 . . . 4 (∀𝑥𝑧 (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On) ↔ (𝜑 → ∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On))
8 rdgon.2 . . . . . . . . 9 (𝜑𝐴 ∈ On)
9 fvres 5198 . . . . . . . . . . . . . 14 (𝑥𝑧 → ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) = (rec(𝐹, 𝐴)‘𝑥))
109eleq1d 2106 . . . . . . . . . . . . 13 (𝑥𝑧 → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
1110adantl 262 . . . . . . . . . . . 12 ((𝜑𝑥𝑧) → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
12 rdgon.3 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ On (𝐹𝑥) ∈ On)
13 fveq2 5178 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
1413eleq1d 2106 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → ((𝐹𝑥) ∈ On ↔ (𝐹𝑤) ∈ On))
1514cbvralv 2533 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ On (𝐹𝑥) ∈ On ↔ ∀𝑤 ∈ On (𝐹𝑤) ∈ On)
1612, 15sylib 127 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑤 ∈ On (𝐹𝑤) ∈ On)
17 fveq2 5178 . . . . . . . . . . . . . . . 16 (𝑤 = ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) → (𝐹𝑤) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)))
1817eleq1d 2106 . . . . . . . . . . . . . . 15 (𝑤 = ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) → ((𝐹𝑤) ∈ On ↔ (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
1918rspcv 2652 . . . . . . . . . . . . . 14 (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (∀𝑤 ∈ On (𝐹𝑤) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
2016, 19syl5com 26 . . . . . . . . . . . . 13 (𝜑 → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
2120adantr 261 . . . . . . . . . . . 12 ((𝜑𝑥𝑧) → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
2211, 21sylbird 159 . . . . . . . . . . 11 ((𝜑𝑥𝑧) → ((rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
2322ralimdva 2387 . . . . . . . . . 10 (𝜑 → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → ∀𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
24 vex 2560 . . . . . . . . . . 11 𝑧 ∈ V
25 iunon 5899 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ ∀𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On) → 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On)
2624, 25mpan 400 . . . . . . . . . 10 (∀𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On → 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On)
2723, 26syl6 29 . . . . . . . . 9 (𝜑 → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
28 onun2 4216 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On) → (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On)
298, 27, 28syl6an 1323 . . . . . . . 8 (𝜑 → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
3029adantr 261 . . . . . . 7 ((𝜑𝑧 ∈ On) → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
31 rdgon.1 . . . . . . . . . 10 (𝜑𝐹 Fn V)
3231, 8jca 290 . . . . . . . . 9 (𝜑 → (𝐹 Fn V ∧ 𝐴 ∈ On))
33 rdgivallem 5968 . . . . . . . . . 10 ((𝐹 Fn V ∧ 𝐴 ∈ On ∧ 𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
34333expa 1104 . . . . . . . . 9 (((𝐹 Fn V ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
3532, 34sylan 267 . . . . . . . 8 ((𝜑𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
3635eleq1d 2106 . . . . . . 7 ((𝜑𝑧 ∈ On) → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
3730, 36sylibrd 158 . . . . . 6 ((𝜑𝑧 ∈ On) → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (rec(𝐹, 𝐴)‘𝑧) ∈ On))
3837expcom 109 . . . . 5 (𝑧 ∈ On → (𝜑 → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
3938a2d 23 . . . 4 (𝑧 ∈ On → ((𝜑 → ∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On) → (𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
407, 39syl5bi 141 . . 3 (𝑧 ∈ On → (∀𝑥𝑧 (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On) → (𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
413, 6, 40tfis3 4309 . 2 (𝐵 ∈ On → (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ∈ On))
4241impcom 116 1 ((𝜑𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306  Vcvv 2557   ∪ cun 2915  ∪ ciun 3657  Oncon0 4100   ↾ cres 4347   Fn wfn 4897  ‘cfv 4902  reccrdg 5956 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-irdg 5957 This theorem is referenced by:  oacl  6040  omcl  6041  oeicl  6042
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